Nope. You get your name changed legally, but not on the BC.

There have been cases of intersex people getting their birth certificate amended, but usually after a long court decision, and not guaranteed.

Yes. If you are adopted, they issue a new one with the names of the new parents on it.

It’s just a piece of paper.
If you know the right people, and you have some money to spread around, I bet you can change anything.

There can be changes made, through the legal system. It isn’t something done from passing fancy.

If you are adopted, as @jca points out, and also if you are born from a surrogate in some states she is on the first birth certificate andbthen it is changed to the parents. In other states the parents go immediately on the birth certificate.

I would assume if there is no father of record and late the father is found his name can be added? I’m not sure how that works.

Here is one link to prove my answer:

http://www.nyc.gov/html/doh/html/services/vr-birth-correct.shtml

I know first hand your answer is correct, @jca! I’d forgotten about that.

thank you @Dutchess_III. I wanted to show those who think it cannot be changed that they are incorrect.

I was teasing Italian Princess on facebook, because she’s finding out what a hassle it is to change her name. She said she went to the DMV and they told her she had to change her SS first. I said, “And THEY’LL tell you have to change your birth certificate!”
She thought I was serious! :D Thus, this question.